Local-to-Global Principles in Floer Theory Umut Varolgunes Stanford - PowerPoint PPT Presentation

Local-to-Global Principles in Floer Theory Umut Varolgunes Stanford University November 14, 2019 Umut Varolgunes Relative Floer theory

Hamiltonian Floer theory Umut Varolgunes Relative Floer theory

Hamiltonian Floer theory ( M , ω ) symplectic manifold. ( L , J L ) tautologically unobstructed Lagrangian. Assume that they are closed for now. Umut Varolgunes Relative Floer theory

Hamiltonian Floer theory ( M , ω ) symplectic manifold. ( L , J L ) tautologically unobstructed Lagrangian. Assume that they are closed for now. Given non-degenerate S 1 / I -dependent Hamiltonian H , we obtain chain complexes over Λ ≥ 0 = Q [[ T R ]]: CF ( H , Λ ≥ 0 ) / CF ( L , J L , H , Λ ≥ 0 ) (1) generated by 1-periodic orbits / 1-chords on L 1 differential counts Floer solutions with weights T topE ( u ) , where 2 � � � u ∗ ω + topE ( u ) = Hdt − Hdt ≥ 0 out in Umut Varolgunes Relative Floer theory

Acceleration data Umut Varolgunes Relative Floer theory

Acceleration data Acceleration data for compact K ⊂ M is a family of time dependent ( S 1 or I ) Hamiltonians H s , s ∈ [1 , ∞ ) such that: H s ( t , x ) < 0, for every t , s and x ∈ K . 1 � 0 , x ∈ K , H s ( t , x ) − − − − → ∈ K , for every t 2 s → + ∞ + ∞ , x / H s ( t , x ) ≥ H s ′ ( t , x ), whenever s ≥ s ′ 3 For n ∈ N , the flow of H n satisfies non-degeneracy 4 Umut Varolgunes Relative Floer theory

Acceleration data Acceleration data for compact K ⊂ M is a family of time dependent ( S 1 or I ) Hamiltonians H s , s ∈ [1 , ∞ ) such that: H s ( t , x ) < 0, for every t , s and x ∈ K . 1 � 0 , x ∈ K , H s ( t , x ) − − − − → ∈ K , for every t 2 s → + ∞ + ∞ , x / H s ( t , x ) ≥ H s ′ ( t , x ), whenever s ≥ s ′ 3 For n ∈ N , the flow of H n satisfies non-degeneracy 4 C ( H s ) := CF ( H 1 , Λ ≥ 0 ) → CF ∗ ( H 2 , Λ ≥ 0 ) → . . . C ( L , H s ) := CF ( L , H 1 , Λ ≥ 0 ) → CF ∗ ( L , H 2 , Λ ≥ 0 ) → . . . Umut Varolgunes Relative Floer theory

Acceleration data Acceleration data for compact K ⊂ M is a family of time dependent ( S 1 or I ) Hamiltonians H s , s ∈ [1 , ∞ ) such that: H s ( t , x ) < 0, for every t , s and x ∈ K . 1 � 0 , x ∈ K , H s ( t , x ) − − − − → ∈ K , for every t 2 s → + ∞ + ∞ , x / H s ( t , x ) ≥ H s ′ ( t , x ), whenever s ≥ s ′ 3 For n ∈ N , the flow of H n satisfies non-degeneracy 4 C ( H s ) := CF ( H 1 , Λ ≥ 0 ) → CF ∗ ( H 2 , Λ ≥ 0 ) → . . . C ( L , H s ) := CF ( L , H 1 , Λ ≥ 0 ) → CF ∗ ( L , H 2 , Λ ≥ 0 ) → . . . The maps are given by continuation maps. Monotonicity requirement (3) implies that topological energies are all non-negative. These are “1-ray diagrams” over Λ ≥ 0 . Umut Varolgunes Relative Floer theory

Definition of the invariants Umut Varolgunes Relative Floer theory

Definition of the invariants We use the telescope construction of Abouzaid-Seidel as a convenient model for homotopy colimits of 1-rays. Umut Varolgunes Relative Floer theory

Definition of the invariants We use the telescope construction of Abouzaid-Seidel as a convenient model for homotopy colimits of 1-rays. Completion functor for modules over Λ ≥ 0 : A �→ � A := lim A ⊗ Λ ≥ 0 Λ ≥ 0 / Λ ≥ r ← − − r ≥ 0 Umut Varolgunes Relative Floer theory

Definition of the invariants We use the telescope construction of Abouzaid-Seidel as a convenient model for homotopy colimits of 1-rays. Completion functor for modules over Λ ≥ 0 : A �→ � A := lim A ⊗ Λ ≥ 0 Λ ≥ 0 / Λ ≥ r ← − − r ≥ 0 SC M ( K , H s ) := � tel ( C ( H s )) (degree-wise completion, whatever the grading is) Umut Varolgunes Relative Floer theory

Definition of the invariants We use the telescope construction of Abouzaid-Seidel as a convenient model for homotopy colimits of 1-rays. Completion functor for modules over Λ ≥ 0 : A �→ � A := lim A ⊗ Λ ≥ 0 Λ ≥ 0 / Λ ≥ r ← − − r ≥ 0 SC M ( K , H s ) := � tel ( C ( H s )) (degree-wise completion, whatever the grading is) LC M ( K , H s ; L ) := � tel ( C ( L , H s )) Umut Varolgunes Relative Floer theory

Definition of the invariants We use the telescope construction of Abouzaid-Seidel as a convenient model for homotopy colimits of 1-rays. Completion functor for modules over Λ ≥ 0 : A �→ � A := lim A ⊗ Λ ≥ 0 Λ ≥ 0 / Λ ≥ r ← − − r ≥ 0 SC M ( K , H s ) := � tel ( C ( H s )) (degree-wise completion, whatever the grading is) LC M ( K , H s ; L ) := � tel ( C ( L , H s )) Homologies are “independent of choices”: SH M ( K ) / LH M ( K ; L ) Umut Varolgunes Relative Floer theory

Definition of the invariants We use the telescope construction of Abouzaid-Seidel as a convenient model for homotopy colimits of 1-rays. Completion functor for modules over Λ ≥ 0 : A �→ � A := lim A ⊗ Λ ≥ 0 Λ ≥ 0 / Λ ≥ r ← − − r ≥ 0 SC M ( K , H s ) := � tel ( C ( H s )) (degree-wise completion, whatever the grading is) LC M ( K , H s ; L ) := � tel ( C ( L , H s )) Homologies are “independent of choices”: SH M ( K ) / LH M ( K ; L ) Automatically get restriction maps for K ⊂ K ′ with the presheaf property Umut Varolgunes Relative Floer theory

Remarks Umut Varolgunes Relative Floer theory

Remarks The main point for well-definedness: any Floer theoretic diagram of chain complexes over Λ ≥ 0 can be “filled” to a homotopy coherent Floer theoretic diagram (only monotone choices are allowed). Umut Varolgunes Relative Floer theory

Remarks The main point for well-definedness: any Floer theoretic diagram of chain complexes over Λ ≥ 0 can be “filled” to a homotopy coherent Floer theoretic diagram (only monotone choices are allowed). Basically a version of Floer-Hofer’s symplectic cohomology (the original one) Umut Varolgunes Relative Floer theory

Remarks The main point for well-definedness: any Floer theoretic diagram of chain complexes over Λ ≥ 0 can be “filled” to a homotopy coherent Floer theoretic diagram (only monotone choices are allowed). Basically a version of Floer-Hofer’s symplectic cohomology (the original one) Similar invariants by Seidel (“Speculations on pair-of-pants decompositions”), Groman, McLean, Venkatesh (maybe Floer?) Umut Varolgunes Relative Floer theory

Remarks The main point for well-definedness: any Floer theoretic diagram of chain complexes over Λ ≥ 0 can be “filled” to a homotopy coherent Floer theoretic diagram (only monotone choices are allowed). Basically a version of Floer-Hofer’s symplectic cohomology (the original one) Similar invariants by Seidel (“Speculations on pair-of-pants decompositions”), Groman, McLean, Venkatesh (maybe Floer?) With M. Abouzaid - Y. Groman: working on extending definition to unobstructed Lagrangians (and their bounding cochains). Significantly harder. Umut Varolgunes Relative Floer theory

Dependence on K Umut Varolgunes Relative Floer theory

Dependence on K SH M ( ∅ ) = LH M ( ∅ ; L ) = 0 (good exercise) Umut Varolgunes Relative Floer theory

Dependence on K SH M ( ∅ ) = LH M ( ∅ ; L ) = 0 (good exercise) SH M ( M ) = H ( M , Λ ≥ 0 ) ⊗ Λ ≥ 0 Λ > 0 Umut Varolgunes Relative Floer theory

Dependence on K SH M ( ∅ ) = LH M ( ∅ ; L ) = 0 (good exercise) SH M ( M ) = H ( M , Λ ≥ 0 ) ⊗ Λ ≥ 0 Λ > 0 LH M ( M ; L ) = HF ( L , Λ ≥ 0 ) ⊗ Λ ≥ 0 Λ > 0 Umut Varolgunes Relative Floer theory

Dependence on K SH M ( ∅ ) = LH M ( ∅ ; L ) = 0 (good exercise) SH M ( M ) = H ( M , Λ ≥ 0 ) ⊗ Λ ≥ 0 Λ > 0 LH M ( M ; L ) = HF ( L , Λ ≥ 0 ) ⊗ Λ ≥ 0 Λ > 0 If K × S 1 ⊂ M × T ∗ S 1 is displaceable from itself by a Hamiltonian diffeomorphism, then SH M ( K ) is torsion. Umut Varolgunes Relative Floer theory

Dependence on K SH M ( ∅ ) = LH M ( ∅ ; L ) = 0 (good exercise) SH M ( M ) = H ( M , Λ ≥ 0 ) ⊗ Λ ≥ 0 Λ > 0 LH M ( M ; L ) = HF ( L , Λ ≥ 0 ) ⊗ Λ ≥ 0 Λ > 0 If K × S 1 ⊂ M × T ∗ S 1 is displaceable from itself by a Hamiltonian diffeomorphism, then SH M ( K ) is torsion. If L is displaceable from K by a Hamiltonian diffeomorphism, then LH M ( K ; L ) is torsion. Umut Varolgunes Relative Floer theory

Dependence on K SH M ( ∅ ) = LH M ( ∅ ; L ) = 0 (good exercise) SH M ( M ) = H ( M , Λ ≥ 0 ) ⊗ Λ ≥ 0 Λ > 0 LH M ( M ; L ) = HF ( L , Λ ≥ 0 ) ⊗ Λ ≥ 0 Λ > 0 If K × S 1 ⊂ M × T ∗ S 1 is displaceable from itself by a Hamiltonian diffeomorphism, then SH M ( K ) is torsion. If L is displaceable from K by a Hamiltonian diffeomorphism, then LH M ( K ; L ) is torsion. Invariance under symplectomorphisms (given by relabeling choices) Umut Varolgunes Relative Floer theory

Descent Umut Varolgunes Relative Floer theory

Descent Let F ( K ) denote SC M ( K , H s ) or LC M ( K , H s ; L ). Umut Varolgunes Relative Floer theory

Descent Let F ( K ) denote SC M ( K , H s ) or LC M ( K , H s ; L ). Floer theory naturally constructs maps F ( K 1 ∪ K 2 ) → cone ( F ( K 1 ) ⊕ F ( K 2 ) → F ( K 1 ∩ K 2 )) (2) We say K 1 and K 2 satisfies descent if this map is a quasi-isomorphism (definition independent of choices). Theorem (V.) If K 1 and K 2 admit barriers (a property independent of L) then, K 1 and K 2 satisfy descent. Umut Varolgunes Relative Floer theory